Development of Criteria for the Design of a Low Noise Level General Aviation Propeller

Appendix A Numerical Analysis of Propellers A-I Appendix B. Former Approach to the Propeller Noise Problem B-I Appendix C Present Approach to the Propeller Noise Problem C-i IV LIST OF SYMBOLS in area in A cross-section E output voltage of a strain gauge bridge ° in in of inertia moment I section in in in inertia moment section polar J K gauge factor of a strain gauge, i e , unit g resistance change per unit strain M bending moment in in-lb Q torque in in-lb R resistance in ohms T thrust load in lb V bridge excitation voltage C strain in microstrains, 1 e , micro-inches per inch EB bending load induced strain "Q torque load induced strain " thrust load induced strain T a stress in psi P Poisson's Ratio Introduction This project, initiated in September 1971, has as its objective the determination of propeller design parameters which yield minimum prdpeller noise, given constraints on propeller performance The study addresses both the theoretical and the experimental aspects of the problem Theoretical efforts have been directed at applying variational techniques to the noise equations Experimental efforts have been concerned with measurement of the propeller performance and noise of a Lockheed YO-3A aircraft The following sections presents a report of the current state of this study 2-1 2 Theoretical Studies The objective of the theoretical aspect of the program is to deter mine the optimum blade loading to minimize noise level, given the thrust, diameter, rotational speed and forward velocity.

Both an aerodynamic model and an acoustic model for the propeller are required for this analysis 2.1 Aerodynamic Model The aerodynamic model at present consists of a vortex system util izing a lifting line and a helical vortex sheet for each blade The blade wise circulation distribution F(y) is the sought for quantity in the noise minimization problem A computer program has been written for the aerodynamic model based, essentially, on the analysis of Moriya CRef. 2 1).

The program, as it now stands assumes constant bladewise section aerodynamic characteristics.

Modification to allow variable section aerodynamics is a relatively easy task and is now in progress The results of the aerodynamic analysis express the propeller performance in terms of the Fourier coefficients for the circulation distri bution F(y) A draft of this analysis is enclosed as Appendix A.

Some further shakedown and check out of the program is required before it becomes fully operational 2.2 Acoustic Model The acoustic model is a simplification of that represented by the Ffowcs-Williams-Hawkings equation (Ref 2.2, Eq 3-16) Presently only the force terms are retained, resulting in a model that is essentially Gutin's model with radially distributed line dipoles representing the blades Following successful treatment of this model, volume displacement effects will be included.

2-2 The sound intensity may be expressed in normalized form, as

I* = F (, r )Y C*)Y(r )dr*dr j

where y is related to the circulation and F is a known function The con straint of constant thrust is expressed as = 1

I y(r*)(l-r*)dr*

Applying variational analysis to these equations provides a necessary condi tion for I* to be a minimum b = - f F(r*,r*)y(r*)dr* 1 1 2 where b is a Lagrange multiplier.

Initial attempts at numerical inversion of this integral equation utilized both a Legendre Polynomial expansion and a Chebyschev polynomial expansion for y(r*), coupled with a Gaussian integration procedure, reducing the integral equation to a matrix equation AL = C where L represents the unknown values of y at selected points on the blade Numerical inversion of this matrix equation was not successful In all cases tried, the coefficient matrix A was ill-conditioned The current approach involves a Fourier transform of the integral equation, implying the assumption that the noise is periodic in the blade frequency, with a Doppler shift This is consistent with the acoustic faT field assumption Some success is indicated with this approach, however convergence is exceedingly slow and, as a consequence, required computer time is large

2-3 Appendix B contains some details of the earlier analysis for a simplified case in which only the thrust component of blade lift was included.

Appendix C contains some details of the current analysis utilizing Fourier Transforms.

2 3 References 2 1 Moriya, T., "Selected Scientific and Technical Papers," University of Tokyo, Tokyo, 1959 2.2 Goldstein, M E , "Aeroacoustics," National Aeronautics and Space Administration, Washington, D C , 1974 3. Experimental Activities All experimental efforts have been directed toward in-flight measure ments of propeller torque and thrust and fly-by sound levels, utilizing a YO-3A aircraft 3 1 Aircraft Preparation Since the YO-3A aircraft was received in disassembled, worn and damaged condition, a good deal of time and effort has been spent rendering it airworthy. Much of this effort was expended in obtaining maintenance and operating manuals, shop drawings, etc. The cooperation in these endeavors of Mr Harold Schuetz, USAAVSCOM, St Louis, and Mr. David Schnebly, Lockheed, Sunnyvale, is very gratefully acknowledged Major repairs have included a top overhaul on the engine, repair of a fractured vertical fin-fuselage attachment fitting, repair of aileron and wing tip damage, replacement of elevator attachment pins, manufacture of special jigs for checking drive shaft/prop shaft alignment and belt ten sion, modification of the forward instrument panel, installation of VHF radio, removal and repair of the muffler system, and other items Photographs of the aircraft and the propeller drive systems are shown in Figures 3 1 and 3.2 Measurement of Propeller Performance 3 2.1 Introduction An essential element in the development of a low-noise propeller is the accurate measurement of its propulsive performance This means that the thrust available, the thrust power available, and the shaft power required must be measured, preferably in flight For the subject studies, a 3-2 Lockheed YO-3A aircraft is available for use as a test platform A signifi cant part of the program has been devoted to instrumenting this aircraft for the in-flight measurement of thrust and power Two approaches to these measurements were considered in this study a) Airflow Measurement By pressure surveys of the flow field behind the propeller, determinations of the thrust and torque acting on the propeller are possible (Ref 3.1) However, this method requires correc tion for the interference effects of the fuselage The accuracy of this method was judged insufficient for the purposes of this proj ect b) Direct Thrust and Torque Measurements The YO-3A's propeller is mounted on a cantilevered pro peller shaft (Figures 3 2 and 3 3) This shaft is driven through a speed reduction drive system employing pulleys and belts Because of this drive system, the engine is isolated from the shaft and load cell measurements on the engine mounts would not yield propeller thrust data Such measurements could produce approximate torque measurements if pulley, belt and propeller shaft bearing losses could be determined As shown in Figure 3.3,approximately 9-inches of the pro peller shaft is easily accessible The strains induced in this shaft are direct measures of the thrust and torque transmitted to the propeller Thus, direct instrumentation of the propeller shaft (say, with strain gauges) can, in principle, yield the desired information This approach was selected 3-3 There is one major difficulty involved in this approach, i e the thrust related strains are much smaller than the torque re lated strains and are very small in an absolute sense No mechani cal modifications to the shaft (e g weakening the shaft in ten sion) were attempted because of airworthiness questions involved Instead, it was decided to attempt to measure the shaft strains directly 3 2 2 Propeller Shaft Loads The YO-3A aircraft is powered by a Continental IO-360A engine rated at 210 hp at 2800 rpm It drives a 3-bladed, constant-speed propeller through a sheave and belt system that provides a 3 33 1 speed reduction ratio. The blade pitch is controlled by a Woodward governor that supplies pressurized engine oil to the propeller through the hollow propeller shaft Based on data given in Reference 3 2 and assuming no drive system losses, it is estimated that a maximum of 186 hp is available at the pro peller shaft to drive the propeller This, together with a propeller shaft speed of 841 rpm, yields a maximum torque to be transmitted by the shaft of 13,944 in -lbs.

Again based on data from Reference 3 2, it appears that the thrust required in level flight, at sea-level, will be in the range from 250 to 400 lbs. Although performance data are not available for the propeller, the assumption of an efficiency of 50% at the sea-level stalling speed of 61 knots yields an estimated maximum thrust available of about 500 lbs.

The thrust load instrumentation has been designed on the basis of a maximum thrust load of 500 lbs, but can measure thrust loads up to 1000 lbs Besides the torque and thrust loads, the shaft is subject to loads from three other sources 3-4 a) Bending Due to Propeller Weight Bending stresses due to the weight of the propeller are estimated to be due to a 1500 in -lb moment acting about the location of the instrumentation on the shaft b) Internal Oil Pressure The internal oil pressure produces an axial tension stress due to the action against the hub piston. It also produces a hoop stress (tension) that, through Poisson's ratio, induces a com pressive axial stress The net effect is a small axial tension stress.

This net stress is about 28 psi for an oil pressure of 100 psi; this is equivalent to an additional thrust load of about 53 lbs c) Centrifugal Force At a shaft speed of 841 rpm, the outer surfaces of the shaft has an acceleration of about 24 g's The resulting axial stress is negligible compared to the other shaft stresses Table I summarizes the stresses and strains in the propeller shaft due to thrust, torque, and bending Both the centrifugal stresses and the net oil pressure stresses as calculated were negligible, however, their effects will be evaluated during calibration of the instrumentation The ratios of the maximum strains taken from Table I are Tmax :Mmax Qmax , 1 6 6 . 42 4 These ratios show the crux of the instrumentation problem, i e., the very low thrust strains in the presence of much larger bending and torque strains 3-5 3 2 3 Strain Transducer Arrangements The sensor choice was dictated by the need to distinguish between the combined loads present, as well as the need to measure them reliably and accurately The sensor must also function over the range of environmental conditions met in aircraft flight. Strain gauges were chosen as the sensors Most strain gauge work is done in the 50-500 mcrostrain range, within this range it is possible to measure changes of 1 or 2 microstrains However, the thrust strains on the shaft are 8.85 microstrains full scale, see Table I. This low strain level poses problems in the thrust measure ment The torque and bending strain levels are high enough to present no difficulties in reliable measurement by foil strain gauges Figure 3,4 outlines the decision process followed in choosing the method of measurement and selection of sensors Two main avenues of approach were made for the thrust measurement 1) Intensification of the low level thrust strains by mechanical means, using foil strain gauges as sensors.

2) Direct shaft strain measurement of thrust strains Foil strain gauge and semi-conductor strain gauge evaluations were made and compared The semi-conductor gauges are 75 times as sen sitive as foil gauges but have a high inherent temperature effect on sensitivity as compared to foil gauges Wheatstone bridges, with an active strain gauge in each arm were chosen If all four arms experience equal temperature changes, and if all four gauges are perfectly matched then temperature effects are automatically cancelled. Figures 3 5 and 3.6 illustrate the directions of the strains produced by the thrust, bending, and torque loads and ,the bridge arrangements used for measuring them. Each of the strains can be measured independently 3-6 of the other two and, in each case, the bridge output exceeds the output of a single active gauge When the shaft is subjected to thrust, bending and torque, the in duced strains act along certain directions, as shown in Figure 3.5 Thrust and bending loads produce tensile and compressive strains in the axial direction Due to the Poisson effect, axial tensile and compressive strains generate compressive and tensile lateral strains. The shear strains induced by torque cause tensile/compressive strains on mutually perpendicular axes inclined at 450 to the shaft axis, the sum of the components of torque strains in the axial and laterial directions are zero The sums of the thrust and bending strain components, in the torque strain directions, are not zero but their effects on the torque bridges will cancel Figure 3 6 shows the three arrangements of strain gauges used to measure thrust, bending and torque on the shaft, taking advantage of the directional properties just discussed Referring to this figure, we have Thrust, E0 - Kg1- o 4 [(+eTJ V Torque E0 - Kg [4Q V Bending E0 4 [4Kg o 4 [B]V Ignoring second order effects, non-linearities and cross-sens'itivi ties of the gauges, and assuming the gauges are perfectly matched and aligned in the required directions, each of these bridge arrangements reacts only to one type of loading and is insensitive to the other two loads 3.2 4 Mechanical Intensification of Thrust Strain Larger thrust strains can be obtained by adding a load path parallel to the propellant shaft and incorporating in this added path a short, weak link shaft (1) carries J4 an arrangement is shown in Figure3.7. The propeller Such of the support link The parallel load path (made up most of the thrust load the of the thrust load, only a small part the weak link (3)) carries (2) and and the pro of the weak link by the relative strengths amount is determined under the load paths stretch equally However, both load peller shaft short weak path occurs in the of the parallel of the stretching Because most that of the shaft, i e , 3 >> I link, its strain is much greater than the following arrangement yields strain intensifier An analysis of this intensification ratio I 3 ti I E 3 A 3 E I = TY 1 A 3 P 2 + ( 2 (T E3) +j- l 3 1 1 3 where micro-inches per inch s = strain, t = length A = cross-sectional area E = modulus of elasticity in Figure 3.7 and 3 refer to the components as shown The subscripts 1, 2, the tensile in the absence of the intensifier If and 1 is the shaft strain of the support link and weak link is made much less than those strength of the such that the propeller shaft 2. I A3B < 1 3AE 1 1 and t. ABE A--J<<1 32 2 ratio is approximately then the intensification 3 I3

I y

3-8 The system used to study the performance of the strain intensifier is shown in Figure 3.8. The system consisted of two weak link assemblies, 1800 apart, mounted on a full-scale model of the exposed portion of the propeller shaft. The links were fabricated of aluminum and were each instrumented with 4-arm strain gauge bridges. Using the smallest gauges available, a 0.2 inches was achieved, yielding a length ratio, ZI/L , of link length of 40. The thickness and width of the links were 0.010 and 0.400 inches, respectively. The tensile strength ratios were 3 =0.003 AI1 A - 0.001 A22 The test assembly was installed in a lathe (Figure 3.9) and could be loaded simultaneously in torque, bending, and thrust. Loadings up to 40% of full-scale were used. Typical results of the thrust calibration are shown in Figure 3.10, and the overall thrust and the torque interactions with thrust results are summarized in Table II. It is evident that the torque interaction is very large. Over the full-scale range of loads to be met in flight, the weak link system would have an output, due to torque interaction, of 860% and 212% of the full-scale thrust output. (The interaction results of the two links are different due to different arrangements of the gauges on each link. This was done to check the effect of gauge arrangement on the torque interaction.) Because of the large torque interaction effects and because the links required very close machining tolerances and were found to be very difficult to install and adjust satisfactorily, the mechanical strain intensifier was rejected as a practical solution to the thrust measurement problem.

3-9 A second attempt at mechanical intensification was made using a deflec tion sensor manufactured by DSC, Inc In this device, a small cantilevered arm is instrumented with semi-conductor strain gauges to measure the bending strains produced by deflection of the end of the arm The deflection sensi tivity appeared very attractive for the present application, i.e , about 6pV/V/w-inch compared to a shaft strain of 8.85 microstrain for the 500 lb thrust load The sensor's arm has a flexure near its end to reduce the effects of bending moments other than those due to linear deflections It was hoped that this feature also would help reduce the torque and bending interactions in the thrust measurement Tests of several different installation arrange ments showed, however, that the interactions were much worse with this sensor.

Because of this, it was rejected as a possible thrust sensor 3.2 5 Direct Shaft Strain Measurements With the abandonment of mechanical strain intensification, a study of the practical problems of direct thrust strain measurement was undertaken.

For comparison purposes, both foil and semi-conductor strain gauges were used In the case of the foil gauges (which have low strain sensitivity but good temperature characteristics), both 120 ohm and 350 ohm gauges were used The latter gauges were used to take advantage of the higher excitation voltages possible. The semi-conductor gauges (which have high strain sensi tivity but poor temperature characteristics) were 1000 ohm gauges with gauge factors of about 155 The gauges were mounted on the model propeller shaft and were in stalled in the lathe (see Figure 3 9) and loaded in combined thrust and torque. The torque sensor was made up of 350-ohm foil gauges fabricated as special purpose torque rosettes Typical results are presented in Figures 3.11 and 3.12 The results are summarized in Table II For 3-10 comparison purposes, the expected sensitivity of the foil bridge was E thrust micro-volts/volt/lb T- 023 and the expected sensitivity of the semi-conductor bridge was F v= 1 78 micro-volts/volt/lb thrust From the results of Table II it was concluded that the foil gauges are adequate for the torque measurement However, the higher sensitivity of the semi-conductor gauges was found necessary for reliable measurement of the small thrust strains Since the aircraft will produce a noisy en vironment for the instrumentation system and since the sensor signals will be transmitted from the rotating shaft through slip rings, it is important that the sensor outputs be as large as possible to insure a high signal-to noise ratio The selection of semi-conductor strain gauges for the thrust measure ment was made in spite of their poor temperature sensitivity characteristics.

The resistance of these gauges increases while the gauge factor decreases as the gauge temperature increases These temperature effects can be reduced by placing a suitable resistance (called a span resistor) in series with the bridge This resistor has a negligible resistance change with temperature As the bridge resistance increases (reducing the circuit current), the voltage drop across the span resistor decreases and voltage drop across the bridge increases This increase in bridge (excitation) voltage tends to compensate for the decrease in bridge sensitivity The value of the span resistance can be used to control the temperature range over which compensation is achieved.

The effectiveness of temperature compensation was studied by installing an electric resistance heater inside the model shaft and measuring bridge 3-11 output as a function of bridge temperature (which was monitored by thermistors mounted near the gauges). Some typical results from this study are presented in Figure 3 13. These results show that the uncompensated bridge has a temperature sensitivity of 12.3 pV/V/ F That is (for a bridge initially balanced at 78 OF) at a bridge temperature of 104 OF, the temperature induced output would be about 40% of the full-scale thrust output However with a 915 ohm span resistor, this bridge, at 1040F, will have a temperature induced output of only 5% of the full-scale thrust output. In flight, the propeller shaft temperature will change due to changes in both ambient temperature and oil temperature inside the shaft. Thus, temperature com pensation of the semi-conductor bridges will be mandatory In addition, sur face temperatures will be measured so that corrections in bridge output can be made.

The results presented in Figure 3 14 and Table II, show that there is a large torque interaction bridge output. This interaction possibly can be explained by one of the following arguments Theoretically there should be no torque strain components in the axial and lateral directions, which are the directions in which the thrust gauges are mounted This is true provided that the thrust gauges are exactly aligned on the shaft axes, and that the shaft is isotropic and homogenous In the present case, the ratio of the maximum thrust and torque strains is 1 42 4 Thus, small misalignments of the thrust gauges from the shaft's axial and lateral directions would cause a pickup of torque strain components that might be sizeable compared to the small thrust strains. The term misalignment here is meant to include both the deviation of the gauges from the axial and lateral directions caused in gauge mounting, as well as the deviation of the shaft axis from its theoretical direction.

For the full-scale loads, the thrust and torque interactions per 3-12 degree of gauge misalignment have been calculated to be- of 96 8% A maximum torque strain interaction for of the maximum thrust strain would occur each degree of misalignment of the thrust gauges A maximum thrust strain interaction of 2.23% for of the maximum torque strain would occur each degree of misalignment of the torque gauges Another possible source of this interaction could be the cross effect, but in the sensitivity of the gauges This is normally a very small present case it may be significant because of the 1 42 4 ratio of maximum thrust and torque strains.

For both the foil and semi-conductor thrust bridges, whose results are presented in Table II, the torque induced output was of a sign opposite torque was the same as that to the thrust output. (The sense of the applied which would occur in the aircraft ) When the sense of the torque was reversed, also changed Both the foil and semi the sign of the torque induced strains slightly in an conductor gauges were mounted on a shaft which had buckled earlier test Before the gauges were installed, the shaft was remachined, but the buckling could have caused a curved shaft axis Another test shaft was made and a new set of semi-conductor gauges the same conditions as the were mounted on it When this shaft was tested under output was still older shaft; the torque interaction in the thrust bridge but its maximum value had been reduced to 44 3%. Also, torques present applied in the same sense as that in the aircraft, produced torque induced strains of the same sign as the thrust induced strains This result is opposite to that observed with the older shaft This result tends to support 3-13 misalignment as a cause of the interaction It is very difficult to exactly predict the misalignment effect because four individual gauges are involved However, for both the old and the new shafts, the torque interaction was calibratable and repeatable A bending interaction in the thrust output was also observed. This interaction was much smaller than the torque interaction The bending inter action can also be caused by misalignment, as well as by small differences in the sensitivities of the gauges making up the thrust bridges For the new shaft, the maximum interaction was only 7 05%. This bending interaction also was found to be calibratable and repeatable 3.2 6 Instrumentation Development and In-Flight Systems Description 3 2.6.1 Introduction and Background A block diagram of the thrust/torque measuring system is presented in Figure 3 15. The elements of this system can be grouped into three sub systems, (1) strain measurement components mounted on the propeller shaft, (2) components located In the cockpit, and (3) auxiliary components. By way of background, a short discussion of the development of the strain measurement system is presented first Because of the very low thrust-induced strain levels, a high gain system with a high signal-to-noise ratio is required if the thrust measure ment is to be successful Initially an AC system, with a 1000 Hz carrier frequency, was designed and buil, This system exhibited sufficient sensi tivity and was quite linear However, the zero-strain bridge output could not be nulled, even when the bridge was resistance balanced Furthermore, this zero load minimum bridge output level was sufficient to saturate the first amplification stage. Is is thought that the problem was due to phase un balance between the semi-conductor bridge arms In view of this problem, 3-14 the system was redesigned to use DC excitation throughout The DC system, which will be used for in-flight testing, has been built and is now being checked out on a full scale model of the propeller shaft. With the exception of the data recorder, all components of the instrumentation system receive power from a regulated + i VDC power supply operating from the 400 Hz air craft electrical system 3.2 6.2 Components on the Propeller Shaft The thrust, torque, and bending transducers (4-arm strain gauge bridges) are mounted directly on the propeller shaft The associated electronics are mounted on a printed circuit board that is attached to the front face of the slip-ring assembly (see Figure 3 16 and Section 3 2 6 3 below) Bridge exci tation is by a regulated, IC 5 VDC power supply that is part of the shaft instrumentation package and operates from the 15 VDC source This additional regulation was selected to provide a stable source for the strain transducer circuits The bridge outputs are processed by IC differential amplifiers feeding into operational amplifiers (See Figure 3.17). The amplifier combinations are adjusted to give overall gains of 1000 The amplifiers are mounted in high thermal resistance packages and have active internal temperature con trols giving very tight temperature control This arrangement provides highly stable gain and low temperature drift over a range of ambient tempera 0 0 tures from -55 C to + 125 C.

3 2 6.3 Slip Ring Assembly A sectional view of the slip ring assembly is shown in Figure 3.18 and a photograph of the components, before final assembly, is presented in Figure 3 19. The brush/ring material combination was selected on the basis of 3-15 information given in Reference 3 3 and after some preliminary laboratory checks The objective was a low resistance contact and very low slip ring noise The materials selected were hard brass rings and silver-graphite (50% silver by weight) brushes Two brushes and two internal leads are pro vided for each ring to further improve the contact characteristics.

In order to avoid the complications of a split slip ring assembly, the inside diameter of the assembly is 5 inches, large enough to slip over the flange at the forward end of the propeller shaft (see Figure 3 2) A split spacer is clamped to the shaft, to bring the diameter up to 5 inches Two aircraft quality bearings separate the inner (rotating) and outer (fixed) assemblies. The outer assembly is thus carried by the shaft and is not rigidly attached to the aircraft frame. A soft rotational restraint will be used, minimizing vibration effects.

3 2.6 4 Cockpit Components An amplifier control box and a data recorder are located in the observer's (forward) cockpit The control box contains active filters, having 3db roll-off at about 65 Hz, and output amplifiers for use in adjusting the overall gain of the system (See Figure 3 20) It will also contain auxiliary components discussed in Section 3 2.6 5 below The data recorder is a Gulton Industries, 8-channel, strip chart recorder provided by the University's Aviation Research Laboratory (ARL) where it has been used for in-flight testing. It is powered by a 400 Hz to 60 Hz inverter, also supplied by ARL A bundle of shielded co-axial cables will connect the slip-ring brushes to the amplifier control box 3.2 6 5 Auxiliary Instrumentation In order to correct the semi-conductor bridge output for temperature 3-16 effects, thermistors will be mounted on the shaft surface close to the strain gauges They will be connected to the cockpit control box through the slip rings and the cable Associated electronics will be contained on the printed circuit board on the shaft and in the control box Their output will also be recorded on the 8-channel recorder The apparent thrust load will contain a contribution due to the oil pressure in the shaft This pressure will be monitored by a pressure trans ducer in the hydraulic line between the governor and the propeller shaft.

Its associated electronics also will be in the control box with output recorded on the 8-channel recorder Finally the control box will contain a single stage timer to provide a time reference for the recorder output 3.3 Fly-by Noise Measurements Capability of fly-by noise measurements has been established Measuring equipment consists of a General Radio 1962-9601 microphone with associated pre-amps and power supply, feeding into a NAGRA IV-SJ tape recorder. Analysis of the results is accomplished by use of a General Radio 1564-A Sound and Vibration Analyzer The noise recording and analysis equipment is University supplaed Procedural checks of the noise measuring program have been satis factorily completed.

Weather conditions with sufficiently low ambient noise level to allow detection of the very low noise level of the YO-3A aircraft have not prevailed at this writing This aspect of the program is ongoing and air craft noise measurements will hopefully be achieved in the near future 3-17 3.4 References 3.1 Vogley, A. W., "Climb and High-Speed Tests of a Curtis No. 714-1C2-12 Four-Blade Propeller on the Republic P-47C Airplane", NACA ACR No. L4L07, 1944.

3.2 Condon, G. IV., Rundgren, I. W., and Davis, W. B., "Army Preliminary Evaluation I, YO-3A Airplane, Final Report", US Army Aviation Systems Test Activity, August 1970.

3.3 Dutee, Phillips, and Kemp, "Operating Stresses in Aircraft Engine Crankshafts and Connecting Rods Part I, Slip Ring and Brush Combinations for Dynamic Strain Measurements", NACA MR E5C0.

3-18 3.5 Bibliography 3.1 Dean M , and Douglas R , "Semi-Conductor and Conventional Strain Gauges," Academic Press, New York and London 3 2 Murray II., and Stein P , "Strain Gauge Techniques," MIT Press 3.3 Perry C , and Lissner H , "The Strain Gauge Primer," McGraw Hill.

3 4 Stein P , "Measurement Engineering," Arizona State University Press

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and Determine Thrust Load Acting Torque on the Propeller Direct Aircraft Flow Field Measurement Structural Survey Behind Propeller Propeller Shaft Pilot Tube and Strain Measurement Vorticity Meters ---- in Slip Stream Strain Intensification Strain Gauges the Shaft Directly on Shaft off Strain Mechancal Strain Mechanical Foil Strain] Semi-conductor Intensifier with Gauges Intensifier with Gauges Strain Strain Foil Strains Semi Cond Gauges Gauges Measurement of Propeller Performance Figure 3.4 Approaches to the In-Flight

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Intensifier I Figure 3.8 Mechanical Thrust Strain Intensifier, Mounted on Model Propeler Shaft "Up" Component of Torque Load; Cable and Pulley

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at Zero Torque C, Loading

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spiral on 0.220 in. wide (typical arranged in a with 0.060 in. spacing) the Outer Tube) Front Retainer Ring---of 13 Lead Connecting Typical Ring to Terminal Post < Rear Bearing Tube (8in. OD.)' 1, Outer Terminal Post (typical of 26) Propeller Shaft Flange (4 7/8 in. O.D.).....

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Fl

~Figure 3.19 The Slip Ring Assembly

Stiess Pei Stiain Per Unlit Load Unit Load Section Strength Strain Ma\imum Strain & Maximum Stress of Section Equation & Maximum Load Characteristics T T -2 = Thrust 529 in -- T T AE= A =1 89 in T T micro-strains/lb AE 5 68x107 lb lb T = 500 = 265 psi =885 Ma\, micro-strains QR Q 0268 Torque micro-strains/ 573 in JQ='2 06 in7 2J Q in-lb 2 2x10 M'Iax = 8020 psi in -lb eQMax = 375aQla\ 14,000 in-lb micro-strains B i103in " 8 IB 309l0 7 MR I-= 0382 -- = 1 14 in M = M micro-strains/ - e B 2_ M2 = i in-lb ln-lb = iax 1500 an-lb a = 8Sa =11 S psi oBf\ 1718 MMaX 58 bearing end (at micro-strains of the shaft) TABLE I PROPELLER SHAFT LOADS

(3o

% Interaction I Pull Scale cMax 100 Scale Full ,T 100 Sensor Remarks Error and QMax TMax de/dQ Max dE/dT krrangement 4eoh Strain Intensifier 840% Hysterisis 630 pV/V .045 IV/V/:n-lb 75 vV/V .15 pV/V/lb Repeatability Fair to pre-load Sensitive Large Torque Inter action Mech Strain I Intensifier I 212% Hysterisis 266 PV/V 019 IV/V/in-lb wV/V/lb 125 wV/V II 25 Fair Repeatability pre-load Sensitive to Large Torque Inter action Bridge Foil Low signal-to-noise

11 34 pV/V 2 94%

00081 PV/V/in-lb 12 0 wV/V 024 IV/V/lb1 on Shaft Hysterisas probler No Temperature

Interaction

-Torque Bridge on High signal-to-noise Shaft 1.56 vV/V/Ib, 780 OV/V 0733 pV/V/in-lb 1.027 mV/V 132% Goore Ineation Interaction ~Torque , Temperature problems _ i micro 12 micro 1680 micro- Torque ' 40 strains strains/ strains Good repeatability Bridge on signal-to-noise < 2 5% High 840 pV/V I in-lb -- 20 wV/V Shaft thrust I Negligible No interaction

1 •temperature problems

TABLE II TEST RESULTS SENSOR EVALUATION

A-I APPENDIX A NUMERICAL ANALYSIS OF PROPELLERS by R. M. Plencner A-2 LIST OF SYMBOLS interference factor a axial factor a' rotational interference A coefficients of sine series n AR aspect ratio B number of blades c non-dimensional chord, C/R C chord, feet CL lift coefficient CD drag coefficient D diameter, feet I induction factor advance ration, V/nD J loading factor k n rotational speed, rev/sec torque, ft-lbs

Q

torque coefficient QC r distance along blade, feet R radius of propeller, feet from origin to segment of trailing vortex filament Tvector T vector from origin to a blade section T thrust, lbs T thrust coefficient c v velocity induced at the propeller disk, ft/sec V forward velocity of propeller, ft/sec A-3 W nnormal induced velocity IV vector sum of rotational and forward velocities X non-dimensional distance along blade, r/R Z rearward distance from the propeller, feet a absolute geometric angle of attack g * induced angle of attack * effective or sectional angle of attack S blade angle F circulation 6interval around the singularity Tefficiency e parametric variable of the helix, radians X tip speed ratio, V/9R slugs/ft density, p a propeller solidity, BC/2r angle between the relative velocity and the horizontal plane rotational speed, rad/sec A-4 INfRODUCTION I.

A great deal of effort and expertise went into modeling the aero dynamic characteristics of propellers in the early part of this century, However, with the coming of the 3et age in the early,1950's, the pro peller was made obsolete for many applications and new theoretical work on propeller theories was almost non-existent Therefore, most of the work done in this area was undertaken before digital computers were de veloped Since the problem of flow around a propeller is by nature very complex, these early approaches to the problem necessitated many simpli fying assumptions and often made use of graphical techniques to obtain solutions to the problem Recent years have seen a renewed interest in propeller modeling for predicting their performance in a wide variety of applications These applications include more efficient propellers to reduce fuel consumption, for commercial propellers suitable for STOL aircraft, low noise propellers as well as rotors for wind mills. Many of and general aviation aircraft, of the applications require more accurate prediction over a wider range with classical propeller theories operating conditions than is possible The purpose of this paper is to review the classical approaches to the propeller problem and then dispense with as many simplifying assumptions as possible in order to produce a theory that more closely models the actual physical phenomenon A-5 II. CLASSICAL THEORIES Momentum Theor - A propeller produces thrust by increasing the velocity of a large quantity of air. This production of thrust is associated with a loss of energy which is due to the increase in kinetic energy, rotational motion imparted by the torque and frictional losses of the propeller blades In the simplest form of the momentum theory, first set forth by R E Froude [1] and W. J. Rankine [2], only the axial component of momentum is considered; the rotational and frictional losses are ignored. Therefore, the theory describes the characteristics of in ideal propeller.

The axial momentum theory is developed [3, 4] by replacing the propeller with a propeller disk that has an infinite number of blades, and the thrust uniformly distributed over the disk. The flow is assumed to be incompressible and irrotational in front of and behind the propeller disk. Under these conditions the change in pressure across the propeller disk is equal to the total pressure head in front of the disk minus the total pressure head behind the disk.

It can then be shown that at the propeller disk the increase in velocity over the free stream value is one half of the increase in the ultimate wake. The efficiency which is predicted by the momentum theory, which, is the maximum theoretical efficiency, is given by V ) = V+ efficiency n = propeller where V is the free stream velocity and v is the velocity increase at the propeller disk.

A-6 Practical applications of the momentum theory are limited due to the gross simplifications that are made The theory neglects rotational energy losses, non uniform thrust loading losses, profile drag, blade interference, and compressible drag changes Its direct application, therefore, is only principally useful in obtaining an upper limit on efficiency or a crude first estimate for the performance of a propeller.

A more general form of the momentum theory [5] includes rotational momentum as well as axial momentum. However, this added complexity does not increase its usefulness in most applications.

Blade Element Theory - The blade element theory [5] analyzes the aerodynamic characteristics of the propeller by estimating the aero dynamic forces experienced by each blade element as it moves through the air. The blade element theory is evolved by assuming that the aerodynamic forces acting on a given blade element are equivalent to the forces acting on a suitable finite wing of the same airfoil section moving linearly with the same relative velocity and the same geometric angle of attack that is experienced by the propeller blade section.

The contributions from all the blade elements may be added up using a strip integration technique to get the overall propeller characteristics.

From the geometry of Figure 1 it can be shown that the incremental thrust and torque of any blade element operating at a geometric angle of attack, ag= - 4, are given by g drf = BCpiV2 (CL cos 4'- C sin4') (2) and d BC rpW (C sin

4 + C cos 4) (3)

dr 2 BLL si +C / / AT /V //

r 1aR

Figure 1. Force and velocity diagram for the blade element theory A-8 where W =V2 + 2 r2 he validity of equations (2) and (3) is largely dependent on the characteristics of the wing used to represent the propeller blade section In order to accurately account for the inter ference effects, the characteristics of the assumed wing would have to change from station to station However, the blade element theory greatly simplifies the problem by assuming the wings used to represent the blade element sections have an aspect ratio of six. Thus the problem is completely specified by equations (2) and (3) when the chord and blade angle distribution and the aerodynamic characteristics of an aspect ratio six wing of the same airfoil section as the blade element are known.

The blade element theory better takes account the effect of the geometric shape of the propeller than does the momentum theory. However, it still fails to accurately account for the interference velocities induced by the trailing vortex system.

Vortex Theory - The vortex theory of propellers is basically a com bination of the momentum theory and the blade element theory. Therefore, it is often referred to as the blade element-momentum theory, To avoid confusion with other theories it will be referred to as the Glauert vortex theory [5, 6, 7] in this paper. The Glauert theory is based on the assumption that the trailing vortex filaments which are produced by the rotating blades form helical vortex sheets as they pass downstream. An exact application would require the induced velocity produced by the trailing vortex system to be computed at each blade station. However, Glauert simplified the problem by assuming the propeller to have an infinite number of blades. This assumption removes the periodicity of the flow, therefore, the velocity for any given radius is constant over disk. As a result, the momentum theory may be used to the propeller evaluate the interference velocities. The geometry of the problem is A-9 given in Figure 2 a and a' are defined as the axial and rotational interference factors respectively V(1 + a) is the forward velocity, V, plus the axial interference velocity. Qr(l - a') is the angular velocity of the propeller blade section minus the rotational interference velocity.

The effective velocity, Wr, of the blade element is just the vector sum of V(1 + a) and Qr(l - a'). From the geometry given in Figure 2 the following relations may be defined

W sin 4 V(1 + a)

(4) W cos = r(l - a') (5) It is often convenient to define a non dimensional speed ratio, A, as the forward speed divided by tip speed. Using equations (4) and (5) the speed ratio may be written in terms of the interference factors as, V 1 -a' _ = I - a tan4) (6) O2R 1 + a The equation for the elemental thrust and torque are found in the same manner as for the blade element theory, except now they are written in terms of the interference factors and are given in non-dimensional form as, Tc r

R d a (1 - a') (C cos - C sin 4) sec 4 (7)

L D and )2 dQc r 4 R _ (1- a') (C sin + C cos ) sec 4 (8) dr LR where the non dimensional thrust and torque coefficients are defined by T = T i R P 2 (9) c and

Q = Q 7R £9

(10) respectively Equations (6), (7), and (8) are not sufficient to determine /.

/

JT

°O

+

vOI+ ci.)

Glauert vortex theory Figure 2. Force and velocity diagram for 1A H A-li the interference factors a and a'. It is now necessary to make use of the assumption of an infinite number of blades, This makes it possible to apply the momentum theory to obtain relations needed to evaluate a and a'.

The results of the axial momentum theory, where rotational momentum is neglected, can be used to determine the axial interference factor as

a a(CL Cos (b - CD in 4)

1 + a 4 sin (I Equation (11) is not exact. In addition to neglecting the flow periodicity due to a finite number of blades, it also is based on the assumption of The ro stream contraction, and no rotational velocity component.

no slip torque to the evaluated by equating the factor may be tational interference terms of angular momentum. This relatlon may be written in rate of increase of the rotational interference factor as, a' cf(C sin + C cos (2 L D (2 cos sin - a' Equations (6), (7), (8), (11), and (12) are sufficient to determine the thrust and torque of a given propeller when the number of blades, the chord distribution, blade angle variation and the aerodynamic charac teristics of the airfoil used at each blade location are known.

Due to the assumptions made, the above set of equations accurately apply only to a lightly loaded propeller with a large number of blades.

However, reasonable results may be obtained for many conditions varying greatly from the above restrictions The predicted thrust and torque distributions along the blade differ from the actual distributions for propellers operating at high blade loadings or for propellers with square tipped blades In the case of square tipped blades, the vortex theory falls to predict the fact that the thrust must fall to zero at the tip [8].

A-12 Goldstein's Theory - Goldstein [9] dispenses with the assumption that the spacing between successive trailing vortex sheets is small and assumes there is a bound vortex filament springing from each blade element The local strength of the trailing sheet is the negative of the circulation gradient around the corresponding blade section The circulation must fall to zero at the root and the tip These trailing vortex filaments approxi mately follow a helical path in the slip stream All the filaments together form a helicoidal surface. Goldstein specifically considers the optimum circulation distribution for which the energy lost due to the production of the trailing vortex system is a minimum, for a given thrust. Betz proved for lightly loaded propellers that this optimum circulation distribution corresponds to the requirement that the vortex sheet move rearward as if it were a rigid helicoid. Thus, the flow in the wake between the helicoidal surfaces is that of an inviscid, continuous, irrotational fluid, with zero circulation The circulation around a given blade element is equal to the discontinuity in the velocity potential of the respective point on the helicoidal vortex surface Since the problem is one of potential flow, it is possible to apply Laplace's equation, v 4 = 0 (13)

where 4 is the

velocity potential. Using the relations = Qz (14) and 2 Vr (15) A-13 Laplace's equation in polar coordinates may be transformed to a more convenient form given by,

+ (I + 21 n- +

= 0 (16)

2 P2 The boundary condition for the problem becomes, S 92rT wV DC V2+02r2 (17) for = 0 or n and 0 <r <R. In addition, 4 must be continuous every where except at the helicoidal vortex surface, and its derivative must vanish when r is infinite. Goldstein goes through a rigorous solution of this problem to obtain the loading factor, k, which is just a non dimensional circulation defined by 1'B k = 2 r t (18) 2w 7r r t an 4 Lock [10] applied Goldstein's solution to a method for predicting the characteristics of an arbitrary propeller. The development of this method parallels that of the Glauert vortex theory quite closely, except that Goldstein's solutions are used to determine the interference velocities at the blades The equations that are obtained for the interference factors are 0 aC a Cos2 cos C sin(1 L D 0(19) a cos) L 1-a k sn2 and a' Cs C Cos + CL sn 20) D

1 - a' k 4 sin 4 cos 4)

Equations (19) and (20) differ from equations (7) and (8) of the Glauert vortex theory by the factor cos2% It should be noted that in the limit k A-14 when the number of blades goes to infinity.

k = cos (21) Thus, equations (19) and (20) reduce to the Glauert vortex theory, equations (7) and (8), for the case of an infinite number of blades.

Goldstein's loading factor, k, only applies exactly when there is a particular distribution of circulation along the blade to give optimum loading as defined in Goldstein's original problem. However, Lock shows that equations (19) and (20) make adjustments for tip losses and are generally more accurate than the Glauert vortex theory formulae, especially in the case of a square tipped blade.

Theodorsen's Theory - Theodorsen [11, 12, 13, 14, 15, 16] attacks the same problem of the optimum propeller as was developed by Goldstein. How ever, Theodorsen expanded and extended the results of Goldstein's work Betz's requirement that the trailing vortex sheet move rearward as a rigid vortex sheet for an optimum loaded propeller was proved only for a lightly loaded propeller Theodorsen extended the interpretation of these results loadings.

to heavy apply to Goldstein's original solution also applied only to infinitely light loadings Theodorsen showed that it is possible to extend these results to heavy loadings by including the rearWard displacement velocity of the helicoidal vortex surface and doing all the calculations in the wake infinitely far behind the propeller This does not change the potential problem that must be solved. However, it does change the operating con ditions which a given solution corresponds to. Including the rearward displacement velocity of the vortex surface, instead of assuming it to be negligibly small, takes account of the contraction of the slip stream.

A-IS Theodorsen's method depends on the solution for the propeller loading coefficient, k, which is a function of the distribution of circulation Goldstein's results may be used to evaluate the loading along the blade.

able to obtain a small number coefficient. However, Goldstein was only 1heodorsen used an electrical analogy method of solutions. Iherefore, using the electrical to experimentally measure the loading function By to obtain the loading function for dual rotating analogy, he was able method propellers as well as for single propellers. The drawback of this gives one is limited to the charts and tables which Theodorsen is that for the loading function.

the Prandtl Theory - Lock [17, 18, 19, 20] put forth Application of propeller problem through an application of the a theory which solved the Prandtl theory [21] for a monoplane wing In calculating the induced velocity along the blade, the actual helicoidal trailing vortex sheets are This method replaced with a system of vortices in two-dimensions.

assumes that the flow around the helicoidal vortex sheets can be approxi by the flow around a system of parallel two-dimensional vortex mated is equal to the actual strength of the helical sheets whose strength axis of rotation. The vortices at the same radial distance from the successive sheets is equal to the pitch of the airscrew distance between The Fourier series may then be used to represent any arbitrary distribution require Strict application of this theory would of vorticity with radius to be found in a manner analogous to the coefficients of the Fourier series equations in that employed in wing theory, where a system of simultaneous be solved. However, to simplify the technique, a the coefficients must of the series and is applied to determine the coefficients graphical method applied to account for errors introduced by replacing corrections are then A-16 the helicold vortex sheet by a system of two-dimensional plane laminae The method in its final form is strictly a graphical technique, using charts developed by Lock Moriya Theory - Moriya [22] developed a calculation method for the aerodynamic characteristics of a propeller in which he obtained the induced velocity by introducing an induction factor, I To obtain the induction factor it is assumed that the trailing vortices springing from each blade form a helicoidal surface which extends rearward from the blade to infinity The downwash velocity at each blade element due to the whole vortex system is then calculated by application of the Biot-Savart law The downwash integral becomes singular when calculating the velocity induced at a point where the trailing vortex filament springs from that same point. If the vortex filament were straight instead of helical, it would exhibit the same singular behavior Therefore, Moriya introduced an induction factor, I, which defined as, d w I n (22) ni where d w is the normal velocity induced by a helical vortex filament and n d w is the normal velocity induced from a straight vortex filament At the point where the vortex filament leaves the blade, the induction factor takes on the value of unity Induction factors for various advance ratios were calculated graphically by Moriya and are tabulated in reference [22] A successive approximation scheme to determine the aerodynamic char acteristics of an arbitrary propeller was developed by Morlya In this scheme the circulation around each blade element is calculated corresponding IL A-17 to an equivalent monoplane wing of aspect ratio six Using this calculated circulation distribution and the suitable induction factors, the normal velocity at each station is obtained by graphically integrating the following.

dl f 4 7r R (23) =iR di x - I x dx W An equivalent aspect ratio for each blade section can then be calculated from w C L n L (24) V Tr AR This scheme is repeated by calculating the circulation distribution corresponding to the new equivalent aspect ratio at each blade element This iteration is repeated until the equivalent aspect ratios of two successive approximations are sufficiently close The lift and drag coef ficients are found in the final iteration and are used in the normal strip theory formulas, equations (2) and (3), to find the thrust and torque coefficients It is reported by Moriya that only one or two iterations are needed to obtain a good degree of accuracy A-18 III 1HEORETICAL INVESTIGATION Most of the preceding theories involve many simplifying assumptions to obtain the interference velocities needed to analyze the aerodynamic characteristics of a propeller The equations and method needed to solve for the induced velocities with a minimum of simplifying assumptions in manner a analogous to that of finite wing theory will now be developed The basic assumption is that of a helicoidal trailing vortex sheet which extends rearward from the blade to infinity. The velocity induced by a segment, dZ, of a trailing vortex filament can be calculated at each blade section by means of the Biot-Savart law which is given as, I di x (t-s)

6 v (r) T7 it-sI

(25) where t is the vector from the origin to the blade section at which the induced velocity is to be calculated and s is the vector from the origin to thevortex filament segment (R. For the geometry depicted in Figure 3 the following relations can be written, t R x'% (26) s = R x cos 6 i + Rx sin j + Ra tan R I (27) d= R x sec d [- cos sinei + cos cos e 3 + sin k] (28) The induced velocity at a point xdue to a semi-infinite vortex filament springing from the point x on the blade is found by substituting equations (26), (27) and (28) into equation (25) and then integrating this equation

over e from zero to infinity The normal component of this induced velocity

is given as,

dw = 4x2f xx cos e + [x(8 sin + cos e) -xiX2 d@ (29)

d 4loR 2O 2 2 2 /2 rX2 /21/2d cosOe + X] [ + +x ±x] a [x -2xx' A-IS vortex geometry Figure 3. Helical A-20 The normal induced velocity at the point x" due to the vortex filaments springing from the radius x on all the blades is eQ-x'] cos e + sin k [x(O 2 k]x'+ cos - xx [x B dr dw (30) 3 ' 2 2dO 41 RI.2 k I 4o 2xxV cos 0k + xX2 [A +x [x (B-k) 0+2Tr where 0 = 0 + B The normal induced velocity at some point x- due to all the vortex filaments springing from each of the blades is found by integrating equation (30) over the blade and can be expressed as 1 dl' B [ ([x) dx

z f

[x2-xx cos +

6k]XI+

[x(O sin Ok cos 0,)- n W X 4rR dx

k=l I

2 _ 2213/2[2 X2 1/2 n [x - 2xx" cosk + x + 0 A [ + (31) Equation (31) is quite complex and is difficult to handle since the inner most integral contains a singularity at x = XV that is not shown explicitly.

This singularity can be made explicit by applying the induction factor, I, introduced by Moriya [22] and defined by equation (22) The induction factor is simply the ratio of the normal induced velocity for a helicoidal trailing vortex system to the normal induced velocity for a straight trailing vortex system of the same strength The induced velocity has a component from the vortex sheet springing from each blade The induced velocity for the helicoidal vortex sheet springing from the blade at which the calcula tions are being made becomes infinite at x = x The induced velocity for a straight vortex system also becomes infinite at x = Z The infinite ,velocity in both cases is a result of that portion of the vortex filament in the neighborhood of the point x'. Since the property of the infinity at x = x' is the same in both cases, the ratio of the two induced velocities must be unity at the point x = x'. The contributions to the induction factor at x = x' from all other helicoidal trailing vortices springing from A-21 all other blades must be zero since only the blade at which the induced velocity is being calculated has the property of infinity at x = x-. Ihere fore the induction factor takes on the value of unity at x - x'.

The induction factor, which is a continuous function of X, x and x' only, may be evaluated numerically by computer. When x approaches x- the induction factor becomes hard to numerically determine with accuracy.

However, since the induction factor is a continuous function and equals unity when x = x, the values when x approaches x' may be interpolated to a high degree of accuracy Typical curves of the induction factor for X = 0.3 are shown in Figure The normal induced velocity at a point x' may now be expressed compactly as, [1 dl' Wn( = f) 4R 'x-x (xdx I dx (32) The singularity at x = x' is now shown explicitly in equation (32) The method is now formulated by applying the fundamental equation of finite wing theory, a 9= a 0- (33) g o 1' a is the geometric angle of attack and is defined by g (34) a = - Arctan (2) The effective, or sectional angle of attack, a , is evaluated by applying the Kutta-Joukowski theorem and is evaluated as, 21' + x,2)1/2 a = ( 352 (35) The induced angle of attack, a is given by i w w a= tan - -- (36)

w

Iq A-22 2.0 1.9 .I N, 1 8 1.7 1.6 1 5 1.4 1.3 1.2 1.1 C, 1.0 .9 .8 .7 .6 .4 .3 .2 I 1II I I I I 1 0 .7 .8 .6 .5 .3 .1 .2 x for X =0 factor curves Induction ligiirc 4.

A-23 equation Putting (34), (35), and (36) into equation (33) results in the fundamental integral equation which must be solved for the circulatlon, F The boundary condition for the circulation requires that, P(O) = F(1) = 0 (37) Therefore, we may express the circulation as a sine series given by F(x') = A sin nnx" (38) Using this series, the fundamental integral equation can be expressed as 2ZA sin niTx' En 7A c n n cos n R Idx / x 2 x - 2

2 2 J

1 + x,2)1 2 (2 2 4rr R Q / ) + x- ( a c R n

S - Arctan (X) (39)

x Difficulty now arises in determining the second term in equation (39) since it contains a singularity in the integrand at x = x' This problem is handled by breaking the integral into three separate integrals, r x-6 I 6 (10

SM dx = M dx M dx + Mdx (40)

0 0 '6f where M cos nlx I and 6 is a small but finite value The first and x - X third integrals on the right-hand side-of equation (40) may be evaluated using established numerical techniques The second term, however, requires special consideration It can be shown that the integrand, M, will go to zero +X6 in the limit as 6 goes to zero Thus, M dx will have a finite value.

A method to evaluate this integral for a finite 6 is required This is accomplished by first making a variable substitution using, t =x-X' (41) A-24 The integral can then be written as (42) + x') I dr Cos nT -6 Equation (42) is then expanded in a power series in T It should be noted that the induction factor, I, is a summation of integrals. The induction factor is a continuous function by virtue of the manner in which it was de fined However, each of the integrals making up the induction factor is not continuous and these integrals do in fact contain singularites There fore, it is necessary to expand the induction factor in a power series in a manner which will eliminate these singularities. Since the singularities occur for the helix angle, 0, equal to zero, they may be eliminated by in tegrating the singular integrals from 0 equal c to - and then separately ex panding the integral from 0 equal 0 to c, which contains the singularity.

Once the power series in T is obtained, it is integrated term by term and terms of order T 2and larger are dropped This gives the following approxi mate relation, x -k=1 2( XA)- x-x I dx = 26 cos (nx') M dx + 26 x2)

JJ-6 k~z X'- 6

(I + Zn(pc2(X2 + x,))- nT 2 sin nnx -6£n(6) 2x. - ') (43) where C is a small value and a >> 6. The singular integral can thus be approximated for the finite region of width 6 around the singularity.

In order to evaluate the coefficients, A , of the infinite sine series given in Equation (38), the series is truncated at N terms. The integral Equation (39) is then applied at N locations along the propeller blade.

This gives a system of N equations in the N unknowns This system of A-25 equations is solved for the coefficients, A n, which are used to evaluate the circulation, using Equation (38) The effective angle of attack can then be found using Equation (35). Once the effective angle of attack is found at each designated position along the blade, the lift and drag co efficients can be found directly from the two-dimensional aerodynamic characteristics of the airfoil used at that position on the propeller blade The differential thrust and torque coefficients can be found by applying Equations (2) and (3).

The preceding method has been programmed for use on a digital com puter. An explanation of the program is given in Appendix A and a list ing of the program is given in Appendix 3.

A-26 APPENDIX A Computer Program Explanation The computer program listed in Appendix B was developed following the method proposed in Section III of this report The variables in Table V must be inputs to the program.

Table V Input Variables to the Computer Program AO - Lift curve slope per radian.

BT - Blade pitch angle in degrees which must be specified at N locations along the blade.

C - Non-dimensional chord which must be specified at N locations along the blade.

CDO, CDI, CD2 - Drag coefficients of the airfoil section where the total drag coeffieient, CD = CDO + CDI a + CD2 ca DELl - Interval around the singularity which is not integrated directly.

EL - Tip speed ratio.

N - Number of terms in the sine series NB - Number of blades.

R - Radius of the propeller in feet.

SKPII, SKPI2 - The upper and lower limits, respectively, around each of the N blade locations, between which the induction factor is interpolated WO - Rotational component of tip speed of the propeller in feet per second.

X,W - Languerre-Gauss integration coefficients.

ZTP - N locations along the blade at which the calculations- for the sine series are made A-27 The program uses a 44 point Languerre-Gauss integration scheme [25] to do the integration from zero to infinity in order to evaluate the induction factors.

Induction factors near x' are interpolated using a spline inter polation formula [26] f Mdx and Mdx are found by using a second-order spline integration technique. The spline integration formula is obtained by integrating the spline interpolation formula given in ref erence [26]. The general form of the spline integration formula for the integral of some function Y between x and x is 1 2 x2 Y dx =-C (x -x +- (x -x 2 ) 2 ) (45) + C 3 - +__2 (x Xl ) 2 where the C's are the spline coefficients generated by the spline inter polation routine developed in reference (27]. The spline integration technique has been found to be very accurate since it fits a curve with continuous derivatives through second order to the given function and then integrates the area under this curve. This integration scheme is also used in calculating the total thrust coefficients from the differential thrust coefficients.

A value of 6 as small as possible is desired to give the best approxi mation for Xf+6 M dx.

M

J, x-6 However, making 6 too small causes f M dx and M dx 0 x+6 A-28 to become less accurate due to the integrand M becoming very large. A - value of 6 between 10 and 10-g has been found to give good results.

The matrix equation for the coefficients of the sine series is solved using Gauss elimination. The answer is checked for accuracy by substituting it back into the matrix equation.

COEFFICIENT FUR AN ARBITRARY PROGRAN COMPUTES THE THRUST C THIS IN CHAPTER THREE THE METHOD DEVELOPED PROPELLER DESIGN USING C OF THIS PAPER.

C C C TO THE PROGRAM ARE: C IMPUT VARIAbLES C THE LIFT CURVE SLOPE PER RADIAN C AO - ALONG THE BLADE AT EACH OF THE N LOCATIONS C BT - THE BLADE ANGLE CD2 - SECTIONAL DRAG COEFFICIENTS C COO, CDI, 4ITH THE RADIUS C C - THE CHORD NON-OIMENSIONALIZED THAT IS INTEURATED INTERVAL AROUND THE SINGULARITY C DELI - THE C SEPARATELY EL - THE TIP SPEED RATIO = V/WO*R C C EP - A SMALL NUMBER WITH EP>>OELI OF POINTS IN THE SINE SERIES C N - THE NUMBER NB - THE NUMBER OF PROPELLER BLADES C THE PROPELLER IN FEET C R - THE RADIUS OF INDUCTION BETWEEN WHICH THE - BLADEWISE LOCATIONS C SKPII,SKPI2 FACTOR IS INTERPOLATED C IN RADIANS/SEC C WO - THL ROTATIONAL SPEED X,W - LANUUERRE-GAUSS INTEGRATION POINTS C LOCATIONS AT WHICH NON-DIMENSIONAL BLADEWISE C ZTP - THE N THE SINE SERIES CALCULATIONS ARE MADE FOR C THc C IMPLICIT REAL*8 (A-H,O-Z) IDUM(9 J INTEGER-4 ZTP(81), ZT(8I), XZ(31), Q181), AA(q,8I),SS(5,81), DIMENSION ), ),SI'IT(9 ),S(ol), RINT(9 ),ALPO(9 ),C(9 I QINT(9 ), CCA{9 ,9 ), YI(RI), CA(9 ,9 ), RHS(9 ), AS(9 DTL(11), BETA(Il) 3 IS(81), TIGRL(1L,1I), COMMON/BBB/X(44), W(44) TT, TTP, PI, EL, EP, NB COMMON/CCC/ TO DEGREES CONVERSION C RADIANS RqC=57.295779513082300 PI=3.14159265358979DO INTEGRATION DATA FOR A 4-4 PLINT LANGUERRE-GAUSS C READ IN READ(5,5) (X(L), WIL), L=1,44) > 5 FORMAT (2D30.0) (' WRITE (6,6) (X(L), W(L), L=1,44) 6 FORMAT if It D30.15, 030.15) N= 9 NB=3 EL=O.29S6DO EP=0.3000 z WO=153. 93800 A=0.07D0*RDC R=1. 40D0 CDO=0.03334821DO CD=-0 .00800893D0*ROC CD 2=0. 00063393DO*RDC*-2 C NP IS THE NUMBER OF POINTS USED IN THE SPLINE INTEGRATION C NP MUST BE AT LEAST 4*N SO THAT WHEN INTEGPATING FRON X=O TO C ZTP-DEL OR ZTP+DEL TO 1 THERE WILL ALWAYS BE AT LEAST 4 POINTS C FOR THE SPLINE INTEGRATION.

(SUBROUTINE SLPIZ WILL FAIL WITH C LESS THAN FOUR POINTS NP= (N-I) +1 00 9 J=I,N C READ IN, THE BLADE ANGLE, CHORD, AND CORRESPONO[NG LOCATIONS C ON THE BLADE READ(5,7) BT, C(J), ZTP(J) 7 FORWAT (3FI0.6) BETA(J)=BT/RDC C ALPS IS THE GEO14ETRIC ANGLE OF ATTACK IF (ZTP(J).LE.O.OOOIDO) ALPO(J)= BT/RDC-PI/2.0D0 IF (ZTP(J).GT.O.OOOIDO) ALPO(J)= BF/RDC-DATAN(EL/LTP(J)) WRITE (6,8) ZTP(J), BT, ALPO(J), C(J) 8 FORMAT (' ', 4F20.6) 9 CONTINUE 00 10 I-1,NP 10 ZT(I)=FLOATiI-I)/FLOAT(NP-1) DELI=0.0000GI0 DO 400 J=1,N DO 4 I=I,NP IS(1)=O 4 YI(I)=O.OD0 WRITE (6,11) ZTP(J) 11 FORMAT ('0', 'ZTP ', F8.3) SH=EL**2+ZTP(J)**2 C NEEDED CALCULATES ALL THE INDUCTION FACTORS C THIS SECTION C J) TTP:ZTP( TT=TTP NOPT=2 AND 2 THE SUM OF THE INDUCTION FACTORS FOR 6LAOES I C P12 IS BY (ZT-LTP) C DIVIDED CALCI (P12, NOPT) CALL NOPT=3 P3 IS THE INTEGRAL FROM EP TO, INFINITY OF THE INTEGRAND M C nHERE 13=(ZT-ZTP)*INTEGRAL OF N C CALL CALCI(P3, NOPT) NOPT=1 TTP=ZTP(J) NAB=O NCTR=O * ICTR=O BETWEEN SKPII & SKPI2 C INTERPOLATE VALUES OF I LOCATION ARE READ IN FOR EACH ZTP WHERE SKPII & SKPIZ C READ(5,12) SKPI1, SKPI2 12 FORMAT (2F10.3) WRITE (6,13) SKPII, SKPIZ I CALCULATION BETWEEN ZT=', F7.3, 13 FORMAT ('O',' SKIP DIRECT I 'AND', F7.3) DO 15 I=1,NP TT=ZT(I) FOR oHICH ZT=ZTP C FIND THE VALUE OF THE DO LOOP PARAMETER IF (DABS(TTP-TT).LT.O.003001O) ICTR=I CALL CALCI(XI, NOPT) IF (TT.LT.SKPI1.OR.TT.GT.SK I2) FACTOR IN THE ARRAY YI C STURE THE INDUCTION YI(I)=XI IF (TT.LT.SKPII.OR.TT.GT.SKP[2) FACTOR CALCULATIONS HAVE BEEN SKIPPED C STORE WHICH INDUCTION IF (TT.GE.SKPIU.AND.TT.LE.SKPI2) IS(I)=1 CONTINUE 00 17 I=l,NP ICK=O TT=ZT(I > I IF (ISII).NE.I-AND.TT.GE.SKPIl-O.O7400.AND.TT.LE.SKPI2+o.O74DO I .OR.I.EQ.ICTR) ICK=2 C NAB IS THE NUMBER OF POINTS AT WHICH THE DIRECT CALCULATION C OF THE INDUCTION FACTOR HAS BEEN CALCULATED IF (ICK.EQ.2) NAt3=NAB+I IF (ICK.EQ.2) XZLNAB)=TT C STORE THE COMPUTED INDUCTION FACTORS IN ARRAY IF (ICK.EQ.2.ANC.I.NE.ICTR) Q(NA6)=YI(I) IF (I.EQ.ICTR) ONAB)=I.O00 17 IF (I.EQ.ICTR) NCTR=NAB WRITE (t,21) NAB, ICTR, NCTR 21 FORMAT ('0', 'NAB=', 3110) WRITE (6,19) (XZ(L), Q(L), L=I,NAB) 19 FORMAT (1 ', 2FZO.5) C USE THE SPLINE INTERPOLATION FORMULA TO OBTAIN THE INDUCTION C FACTORS THAT WERE SKIPPED IN THE DIRECT CALCULATIONS C SLPIZ IS A COMPUTER SUPPLIED SUBROUTINE WHICH WHICH GENERATES C SPLINE INTERPOLATION COEFFICIENTS.

CALL SPLIZ (XZ, Q, NAB, AA, SS) DO 18 I=4,NP TT=ZT(I) C USE SPLINE INTERPOLATION FORMULA IF (IS(I].EQ.1.ANO.I.LT.ICTR) I YI(I)=AA(I,NCTR-I)*(XZ(NCTR)-TT)1C3 I +AA(2 ,NCTR-A)*(TT-XZ(NCTR-i) )**3 2 ±AA(3,NCTR-I)*iXLNCTR)-TT) 3 +AA(4,NCTR-1)*(TT-XZ(NCTk-I)) IF(I.EQ.ICTR) YI(I)=1.ODO 18 IF (IS(I).EQ.I.AND.I.GT.ICTR) I Y (1)=AA(I,NCTR)*(XZ(NCTR+I)-TT),C*3 +AA(2,NCTR)*-(TT-XZ(NCTRI)4*3 3 +AA(3,NCTR)*(XZANCTR+L)-TT) 4 +AA(4,NCTR)(TT-XZ(NCTR)) IF ZTP(J).GT.DELL) TT=ZTP(J)-DELI IF (ZTP(J).GT.DELI) 1 XI6L=AA(I,NCTR-I)*LXZ(NCTR)-TT)**3 1 +AA(2,NCTR-I)*(TT-XZ(NCTR-I))**3 +AA(3,NCTR-1)*(XZ(NCTR )-TT) +AA(4,NCTR-I)*(TT-XZ,CrR-1)> ;,z IF (I.OOO-ZTP(J )SGT.DELI) TT=ZTP(J )+DELl IF ( 1.0O00-ZTP(J) .GT.DEL1) I XIAZ=AA( 1,ICTR)*(XZ(NCTR+I)-TT)*'3 2 +AA(2,NCTR)*(TT-XZ(NCTR))**3 3 +AA(3,NCTR )*( XZ(NCTR+1)-TT) 4 +AA(4,NCTR)*(TT-XZ(NCTR)) C ALL THE INDUCTION FACTORS HAVE NO, bEEN CALCULATED AND C STOkED IN YI, XIBZ, XIAZ. XI3Z & XIAL ARE THE INDUCTION C FACTORS AT ZTP-DEL AND ZTP+OEL RESPECTIVELY.

00 210 NS=lN AP G=iS4P I*ZTP (J WRITE (6,20) NS 20 FORMAT (10', '+++ NS=4, 13, ' ........... ) 00 22 IJ=I,N QINT( IJ)=0.O0O RINT(I J)=O.ODO 22 SINT(IJ)=O.ODO DEL=OELI ND=O IF (ZTPeJ).LE.DEL) GO TO 53 C THIS SECTION CALULATES THE INTEGRAL FROM 0 TO ZTP-DEL C D0 25 I=1,NP ND=tD+ 1 IF (ZT(I)-ZTP(J).GE.-DELI ) GO TO 30 XZ(I)=ZT I) C STORE THE INTEGRAND IN ARRAY 0 25 Q(I)=DCOS(NS*PI4ZT(I)*YIU()/(ZT(I)-ZTP(J)) 30 XZ(N0)ZLTPCJ)-DEL w(ND)=DCOS(NS4PI*XZi(ND) )'XL3Z/(XZ(ND)-ZTP(J)) NDI=ND-t IF (NS.EQ.1) WRITE (6,35) (XZ(L), Q(L), YI(L), L= ,ND ) IF (NS.EQ.l) WRITE (6,35) XZ(NU), Q(ND), XIBZ 35 FORMAT (' ', 3015.5) C INTEGRATE Q USING SPLINE INTEGRATION FORMULA CALL SPLIZCXZ, t, NO, AA, SS) SUMQ=O.ODO ND I=ND-I 00 40 K=I,NDI 40 SUMQ=SUMQ+.25DO*AA(I,k)*(XZ(K+L)-XZ(K))**4 1 +.2500*AA(2,K)*(XZK+1-XZ(K))**4.

2 +.5000*AA(3,K)*(XZ(K+I-)-XZ(K)})**2 3 +.50DO*AA(4,K)*(XZ(K+I)-XZ(K))**2 QINT(NS)=SUMQ WRITE (b,50) SUMQ 50 FORMAT ' ', 'INTEGRAL OF Q=', E1I.3) 53 CONTINUE IF (IGT.EQ.I) GO TO 75 IF (ZTP(J).LT.DEL.OR.1.ODO-ZTP(J).LT.DEL) DEL=DEL/2.DO C C THIS SECTION CALULATES THE INTEGRAL FROM ZTP-OEL TO ZTP OEL C ATERM=DCOS( ARG)*2.0OO*OEL*P12 BTERMI=2.0OO*OEL*((ZTP(J)*OCOS(ARG))/OSiRT(2.0DO)*SH) *(1.ODO+DLOG(2.OOO*EP**2*SH)l 2 -NS*PI*DSQRT(2.ODO)*DSIN(ARG)) BTERM2= -DEL* OLOG(OELL)*(4.00*ZTP(J)tOCOS( ARG) 1 /(DSQRT(2.ODO*SH)) CTERM=2.0OO*DCOS(ARG)VDEL*P3 RINT(NS}=ATERM+ TERMN+-BTERM2+CTE M IF (NS.EQ.1) WRITE (6,55) P12, P3 55 FORMAT (10', 2E20.5) IF (NS.EQ.1) WRITE (6,60) ATER.1, BTERMI, 3TERM2, CTERM 60 FORt'AT ('0', 4E20.5) WRITE 16,70) RINT(NS) 70 FORMAT ' , 'INTEGRAL OF R=', E1I.3) DEL=DELI 75 RIaNTINS)=O.O00 IF (1.O00-LTP(J).LE.DEL) GO TO 160 C THIS C SECTION CALULATES THE INTEGRAL FROM ZTP+OEL TO 1.0 C ND=1 XZ(ND)=ZTP(J)+OEL S(NO)=DCOS(NStPI*XL(ND))*XIAZ/(XZ(ND)-ZTP(J)) DO 100 I=I,NP IF CZT(I)-ZTP(J).LE.OELI ) GO TO 100 > IA ND=ND+ XZ(ND)=ZT(I) C STORE THE [NEGRANO IN ARRAY S S(ND)=OCUS(NS*PIZT())4Y1(I1IZT()-ZTp(J) 100 CONTINUE 110 CONTINUE IF (NS.EQ.I) WRITE (c,35) XZ(1), S(1), XIAZ LNO=NP-NO DO 115 L=2,NO LL=L+LND IF (NS.EQ.1) WRITE 16,3D) XZ(L), S(L), YI(LL) L15 CONTINUE C INTGRATE S USING SPLINE INTEGRATION FORMULA CALL SPLIZ(XZ, S, NO, AA, SS) SUMS=O.000 NDL=ND-1 00 140 K=I,ND1 SUMS=SUiS+.2500*AA(I,K)*(XZ(K+I)-XZrK))**4 I +.25O*AA(2,K)*(XZ[K+L)-XZ(K))*4 2 +.50DOAA3,K)*(XZ(K+I|-XL(K))**2 3 +.50CO*AA(4,K)*(XZ(K+1)-XZ(K))4*2 SI NT{NS)=SUMS WRITE(6,150) SUMS = 150 FORMAT (' ', 'INTEGRAL OF S I, E12.4) C TINT IS THE TOTAL INTEGRAL FROM ZERO TO 1 160 TINT=QINT(NS)+RINT(NS)+SINTCNS) TIGRL(J,NS)=TINT WRITE(o,170) TINT 170 FORMAT ('0', 'TOTAL INTEGRAL FROM 0 TO INFINITY=', E20.5) C SET UP THE MATRIX EQUAfION CA(JNS)=2.00*DSIN(NSvPI*ZTP(J))/AO-NS4C(J)*T1NT/&.0D0-\ CCA(J,NS)=CA(JNSJ CONTINUE RHS(J)=WO4_*SQRT(SH)*ALPO(J)*C(J) WRITE (6,350) 350 FORMAAT (101, CONTINUE WRITE(6,500) ((CA(M,L}, L=1,NI, M=1,N) > FORMAT (I t, 9FiL.5) WRITE (6,550) fRHS(L), L=I,N) FORMAT ('0, 9F12.3) C SOLVE THE MATRIX EQUATION FOR THE AS VALUES USING A C COMPUTER SUPIDGAUSS ELIMINATION TECHNIQUE.

CALL GAUSZ (CA, N, N, RHS, AS, OUM, IER) WRITE fb,oOo) (ASCL), L=I,N) FORMAT (I I? 'A=', D20.5) WRITE (b,o25) FORMAT ('', fix, 'ZETA', 16X, 'GAIA', 16X, 'ALPHA I', 1D 14X, 'ALPHA 01, I4X, 'DTC', 14X, 'PHI') D0 NK=1,N ST IGR=O.

0DO SGAM=0.O00 VC=iON*RE-OSQRT( EL**2+ZTP(NKq**2) C VR IS THE RELATIVE VELOCITY VR=VC/R DO K=IN ST IGR ST IGR+TIGRL (NK, K)} 4 (K K C CALULATE THE CIRCULATION o50 SGAP=SGAM+AS(K)* OSIN(KPI*ZTP(NK)) AL I-=ST IGR/(4.ODO*VR) C CALCULATE THE EFFECTIVE ANGLE OF ATTACK ALO=2.

CDOASGAM/ (AOWC (iNK) *VC) PHI=BETA (IWK) -ALO IF (ALO.LE.0.L08O0) CL=O.LIDO-RDC*ALO C CALCULATE THE LIFT COEFFICIENT IF (ALO.GT.o.lOSoO) CL=0.O7OOOROC(ALO-O.O80O)+o.7O0 C CALCULATE THE DRAG COEFFICIENT CD=CDO-C0I- ALO*CD2Z*ALO* CY=CL*OCOS(PHI )-CO*DSIN(PHI) C CALCULATE THE DIFFERENTIAL THRUST OTC(NK)z NB*C(NK)*VR*2*CY/(2-ODO PI*P= 2*RO, } (O.PSOO Pli,3) WRITE(6,675 ZTP(NK), SGAM, ALI, ALO, DTC(NK), PHI FORMAT (' ', 6E20.5) CONTINUE C INTEGRATE THE THRUST OVER THE BLADE USING SPLINE INTEGRATION C FORM1ULA CALL SPLIZ(ZTP, OTC, N, AA, SS) NM I=N- 1 SUMT=O.O O DO 702 K=I,NMI 702 SUMT=SUMT+.25O0DOAA(2,K)*(ZTP(K+I)-ZTP(K}))**4 +-500DO*AA(4,K)*(ZTP(K+I)-ZTP(K))**2 +.250DO*AA(I,K)*(ZTP(K+1)-ZTP(K))**4 e-.5000DXAA(3,K)*(ZTP(K-I)-ZTPh()*}'2 WRITE (6,705) SUMT 705 FORMAT ('0', 'THRUST COEFFICIENT=', F20.7; C CHECK TO SEE IF THE MATRIX EQUATION WAS SOLVED CORRECTLY D0 725 I=I,N CKRHS=0.0 I DO 710 J=1,N 710 CKRHS=CCA(I,J)*AS(J)+CKRHS 725 WRITE (6,730) CKRHS 730 FORMAT ('0', F20.31 RETURN END C C SUBROUTINE CALCI (XI, NOPT) C THIS SUBROUTINE CALCULATES THE INDUCTION FACTORS AS FOLLO4S: C NOPT = I CALCULATES 11+[2+13=1 C NOPT = 2 CALULATES THE INTEGRAL OF PI+P2 FROM 0 TO INFINITY C NOPT ='3 CALULATES THE INTEGRAL OF P3 FROM EPSILON TO INFINITY C WHERE E=(ZT-ZTP)*(PI+P2+P3) IMPLICIT REAL*8 (A-H,O-Z) COMMON/AAA/ KK, NNOPT COMMON/CCC/ TT, TTP, PI, EL, EP, NB DIMENSION ANJ3) EXTERNAL AUX NNOPT=NO(PT DO 10 KK=1,3 IF (NOPT.EQ.2.AND.KK.EQ.3) GO TO 20 IF (NOPT -EQ.3.AND.KK.NE.3) GO TO 10 CALL GL0U (AUX, ANS) 10 AN(KK)=ANS IF (NOPT.EQ.I.AND-DABSTT-TTP).LE.O.0OIDO) I XI=t.0OO -, IF (NOPT.EQ.I.AND.DABITT-TTP).GT.O.OOIDO) 1 XI=(AN[X)+AN(2)+AN(3))*(TT-TTP) 20 IF (NOPT.EQ.2) XI=AN(t)+AN(2) IF (NOPT.EQ.3) XI=AN(3) RETURN END C C SUBROUTINE GLQU (AUX,ANS) C THIS SUBROUTINE SETS UP THE LANGUERRE-GAUSS INTEGkATION IMPLICIT REAL*8 (A-H,O-Z) COMMON/BBB/X(44)y W(44) 200 ANS=O.ODO S=O.ODO 00 201 I=1,44 Y=X(I) CALL AUX(Y,Z) 201 S=S+Z*4(I) ANS=S RETURN END C C SUBROUTINE AUX (PHI, FX) C THIS SUBROUTINE CALCULATES THE INTEGRANO OF Pl, P2, & P3 c WHERE THE INDUCTION FACTOR = (ZT-LTP)*INTGRAL OF (PI, P2, &P3) C NOTE THAT THE INTEGRAND OF P IS GIVEN BY A/B BELOW IMPLICIT REAL*8 (A-HO-Z) COMlfON/CCC/ TT, TTP, P1, EL, EP, NR COMMON/AAA/ KK, NOPT IF (NOPT.EQ.3) PHI=PHL+EP PHIK=PHE+2.0DO*PI*(NB-KK)/N6 '-'=T**)*CTT-TTPDCOS(PHIK))+TT*PHI*DSIN(PHII+DCOSPHIK))-TTP) 1 ;cEL**2 B=(DSQRT(TT**2+TTP*t2-2.0OO*TT*TTP*DCOSIPHIK)+PHI**24EL**2))**3 I *DSQRT(EL-**Z /r *2) C=A/B 4p r7P FX=DEXP(PHI)*C RETURN A-39 REFERENCES 1. Froude, W , "On the elementary relation between pitch, slip and propulsive efficiency," Transaction of the Institute of Naval Architects, Vol. 19, 1878.

2. Rankine, W. J M , "On the mechanical principles of the action of propellers," Transaction of the Institute of Naval architects, Vol.

6, 1865 3. Dommasch, D. 0., "Elements of Propeller and Helicopter Aerodynamics," Pitman, New York, 1953.

4. McCormick, B. W., "Aerodynamics of V/STOL Flight," Academic Press, New York, 1967.

S. Durrand, W F., "Aerodynamic Theory, Vol IV," Julius Springer, Berlin, 1945 6. Glauert, H., "The Elements of Aerofoil and Airscrew Theory," Univer sity Press, Cambridge, 1948 7. Glauert, H , "An Aerodynamic Theory of the Airscrew," -Aeronautical Research Committee Reports and Memoranda No. 786, 1922 8. Glauert, 1f , and Lock, C N. H , "The Accuracy of the Vortex Theory of Airscrews in the Light of Recent Experimental Work and Its Applica tion to Airscrew Design," Aeronautical Research Committee Reports and Memoranda No 1040, 1926.

9. Goldstein, S , "On the vortex theory of screw propellers," Proceedings of the Royal Society of London, Series A, Vol. 123, 1929 10. Lock, C N H , "Application of Goldstein~s Airscrew Theory to Design," Aeronautical Research Committee Reports and Memoranda No 1377, 1930 11 Theodorsen, T , "Theory of Propellers," McGraw-Hill, New York, 1948.

12. Theodorsen, T , "The Theory of Propellers I - Determination of the Circulation Function and Mass Coefficient for Dual-Rotating Propellers," NACA Report 775, 1944.

13. Theodorsen, T , "The Theory of Propellers II - Method for Calculating the Axial Interference Velocity," NACA Report 776, 1944.

14. Theodorsen, T., "The Theory of Propellers III - the Slipstream Con traction with Numerical Vlues for Two-Blade and Four-Blade Propellers," NACA Report 777, 1944 Az40 15. Theodorsen, T , "The Theory of Propellers IV - Thrust Energy and Efficiency Formulas for Single and Dual-Rotating Propellers with Ideal Circulation Distribution," NACA Report 778, 1944.

16. Crigher, J , "Application of Theodorsen's Theory to Propeller Design," NACA Report 924, 1949.

17. Lock, C. N. H , "Application of Prandtl Theory to an Airscrew," Aeronautical Research Committee Reports and Memoranda No.

1521, 1932.

18. Lock, C. N H , "Airscrew Theory," Aeronautical Research Committee Reports and Memoranda No. 1746, 1936.

19. Lock, C N. H.

and Yeatman, D., "Tables for use in an Improved Method of Airscrew Strip Theory Calculation," Aeronautical Research Committee Reports and Memoranda No. 1674, 1934.

20.

Lock, C. N H., "A Graphical Method of Calculating the Performance of an Aarscrew," Aeronautical Research Committee Report and Memoranda No. 1849, 1937.

21. Prandtl, L , "Application of Modern Hydrodynamics to Aeronautics," NACA Report 116, 1921.

22. Moraya, T., "Selected Scientific and Technical Papers," University of Tokyo, Tokyo, 1959 23 Reid, E , "The Influence of Blade-Width Distribution on Propeller Characteristics," NACA Technical Note 1834, 1949 24. Stack, J , "Test of Airfoils Designed to Delay the Compressibility Burble," NACA Report 763, 1943 25. Stoud, A. and Secrest, D , "Gaussian Quadrature Formulas," Prentice- Hall, Englewood Cliffs, New Jersey, 1966.

26. Pennington, R. H , "Introductory Computer Methods and Numerical Analysis," MacMillan Company, London, 1970 27. FORTUDI Writeups, Computing Services Offices, University of Illinois, Urbana, 1974.

B-I APPENDIX B FORMER APPROACH TO THE PROPELLER NOISE PROBLEM by C. J Woan LIST OF MAIN SYMBOLS a speed of sound t observation time o of source the velocity V intensity acoustic propeller of forward velocity Vf per unit length ZCr) thrust = V/a X field point = V/a (X 1 + x02 + x23) 12 the distance fletween the field point and the center M r t2/a of the propeller at t t Y source point

MR M-R/R

e the instantaneous

angle the source P1 acoustic pressure makes with y - axis Pt acoustic pressure due to 00 initial angular displacement thrust T tan = xo2/Xol 'r retarded time T radius of propeller t Q2 angular velocity of propeller T total thrust [f(t)] f(T) System and Geometn' i. Coordinate x Y3 Point Source Lifting line

0 0

(Xl'x2'O) Point Field System Figure 1 Coordinate B-3 2. Acoustic Pressure Perturbation Following Ffowcs Williams and Hawkings, the acoustic pressure perturbation at the field point is given by th where dS is the blade surface area and P is the i component of the force acting on air by the blade surface 3 Noise Due To Thrust Let Z(r) be the spanwise distribution of the thrust along the lifting line per unit length Then at I-R k where P, -Z(r) P2 0 p =0 Therefore the pressure perturbation due to thrust is

E

T f

(-

dr

where

= r * R

rv~ B-4

= r) r (I-n

thrust T is the total and rSi7t CCos

N21 'nkFr 4 ~g~

Mt

"ExMc s rv$ s

r t Cosrost []-- Then the proba where the angle 6 is uniformly distributed over (0,2v) ° bility density function for the random variable 6 is -- 0 otherwise P' is Therefore, the mean square of t The intensity define Then !I

c__r1 - Kr r,)if

)(r,) f dr, 9C )

where

ft ( &,e) def

F% fl ) 3- fp ,o)

by The total thrust is given

) Jr

( -Y

(r)r

T

-T

Letting

(r) r (I-r ) dt)

=

to the constraint necessary condition that I* be an extremum, subject the (i.e. constant total thrust) is J = const

Icrtrj)

(btdK

-

-

(2)

where b is a Lagrangian multiplier and b can be obtained by solving equations (1) and (2) and &*(r*) Numerical Method 4.

to solve this problem are The equations needed

fr

t*I

X~r P*

(A)

b

(B) _ = -T i S* 2.

() Mt r c i-i) r

_nPo Vo(lo 2*~q L(M %0 IfoCoCos~~

(o>~CO

I ti r Mt SnIq

_________ B-6

(E). - fJ 2j (~Cos I' M le) 05T 2t

r254Pmt s 1 rL cos + t

(F) [9] = oo - t

(G) [i- ) = ) (M C cs- - stO , [9) The solution proceeds as follows Given MF, MT, xo, and T, find b and T*(r*) 1 The integrals in equations (A), (B), and (C) are calculated by Gaussian integration (Gauss-Legendre formulas or Gauss- Chebyshev formulas).

2 Equation (E) is solved by the Newton Method f WI)

t = (rRM_,, g.tC-os (0(g 2( -t'- 4

3 After obtaining [R*] from (2), the value of [R*] is substituted into equations (A), (B), (C), (D), (F), and (G) 4. After applying Gaussian integration to equation (B) and assign ing n different values for t* (o.<r<l, n = the number of points used in Gaussian integration), equation (B) can be reduced to a system of n equations with unknowns T*(r*) (i=l,2,--,n) t h [r* are the zeros of the n polynomial used in the Gaussian integration] B-7 In matrix form AL = C A. nxn matrix C* column matrix 5 The Lagrangian multiplier is obtained by substituting Z*(r*) obtained from (4) into equation (A).

C-I APPENDIX C PRESENT APPROACH TO THE PROPELLER NOISE PROBLEM by C. J. Woan LIST OF SYMBOLS A Fourier coefficients of F T total thrust t a speed of sound t observation time B number of propeller blades V the velocity of source C complex magnitude of the V forward velocity of propeller Cm thprple F n harmonic IW. induced velocity dD drag per unit length field point dL lift per unit length

V source point

I~ sourcel poinstt acoustic intensity a induced angle of attack Jm Bessel function of order in m(Z) fntinitial angular displacement and argument Z 0 the instantaneous angle the 1 = V/a source makes with the Y - axis M = M - R/R PO fluid density R circulation distribution r spanwise P-P acoustic pressure T retarded tame P root-mean-square pressure rms Cp angular velocity of propeller R X- Y [f(t)] EfCT) R= 222 +z +y x R0 O r radius of propeller t C-2 1.Coordinate Sys tem and Geometry z x Y3 Source Point Line Lifting (x,y,z) (x ,x , 1 2 0 ) Field Point Figure 1 Coordinate System C-3 Sound Pressure Due to Moving Lifting Line 2.

The force acting on the air at Y = gT+r, by the moving lifting line, is developed as follows Consider a blade element at location r

oiL

F

dKx=-(dL

cos-4r- dDi ,

t dF%

dLsinjr-±dDcost

The force, dF, acting on the air at Y £2T + F is

d f= d F, Z - daFe si-S a T + dPF,,cos0 o

The total sound pressure generated by the moving lifting line is given as

47ra~ Z)

R ( I-M~

Periodicity of the Acoustic Pressure Disturbance 3 Consider a propeller whose hub is moving with constant velocity V.

Choose fixed axes 0 such that the origin coincides with the hub xy z (i) at time T = 0 (ii) the x axis is parallel to the axis of rotation C-4 (iii) the hub velocity vector lies in the plane and in x-direction.

Let the field point f be at (x,y,z) and a typical source point be situated at Y(t) This can be expressed as the vector sum of the hub location and the rotational position

yV() = VF

- - r (-c), r(-) = ( rcose, rsi, Here 0 is the instantaneous angle the source makes with y axis This in creases linearly with time, and can be expressed as QT+ where ? is the angular velocity of the source, and 0 is the initial angular displacement.

[RI

a.

If attention is confined to source times near T=O, [R] can be approximated by

2 j

t{(t-v,.) C R]

+

rcosW})

t ( rsr[0)4 S

where R (t± + ±

and =Lr'

-2r(Wcoscej + Osirnfo+J b~l2 j

X Vmr P10

Therefore,

P-3 P-- -~V CO 03 L 4

and

X1

t - - -(2 r (E Mor) -M r CO[s 3 + HzsRl[ )

C-5 where r- M and

Nor

QbkO = a, 0~ This equation shows that as e increases by 2 , Pt only increases by 2w (l-m or) Hence the basic observed frequency is w=&I/(l-M or ) and therefore P-P has a period 2 /w 4 Fourier Analysis (P-Po) (Xt) has a period 2, where w = Q/(r-Mor) VFx M and or Roa O t h Then, defining the complex matnitude of the m harmonic in the usual way 27C

( p-p )(3

t) e dt

Wher( =) 5 L df'Mc)c+iJ

changing variables back to the retarded time, T, gives

F

Ce j

-w-)

dr

where dt =L-Mg p dr

- [0

( cosC&J + sniLe)

Since the far field approximation is in force, the only significant dependence of FI upon x is through the R term in the exponent.

i I0 C-6 Therefore - o dF.-

]

e and C n=-'i--- ~o~~-tl, 2CM (-'C ) (n -- )" .

2 7[ ao(9, To (I)

(#-7n.

Mtt 5. Extension to B Blades the usual phase arguments show that For an assembly of B identical blades, survive, and all the rest cancel only harmonics which are multiples of B frequency, Bc are Hence only harmonics of the propeller blade passing present in the acoustic field of the complete propeller, and the magnitude is BCmB: / of the mth harmonic M

(I - Mor)(,b

a. R.

2- V

Mt Mt Then, the r m.s pressure is given as at X is, The total intensity

fa

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Tt=5 (r tV V

9 ) d r

V

6. Method of Solution The problem to be solved can be stated as follows Find the circulation distribution along the lifting line such that I is minimum and Tt constant.

The solution then proceeds as follows.

Expand F as Fourier series with coefficients A then

I =

I (A ,, A,, A

. .) = T (Ak)

Tt = -t ( At, A , A • T A)

2 3 Let J = I - b(Tt - Tt (A) ) where b is a Lagrangian multiplier.

A necessary condition that I be extremum is -=

i

2 -2, 3, Solve equation (A) and (B) for AI and b